Nonlinear Lie higher derivations on triangular algebras

Abstract

Let ℛ be a commutative ring with identity, A, B be unital algebras over ℛ and M be a unital (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let be the triangular algebra consisting of A, B and M. Motivated by the powerful works of Brešar [M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), pp. 525–546] and Yu et al. [W.-Y. Yu and J.-H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010), pp. 2953–2960], we will study nonlinear Lie higher derivations on 𝒯 in this article. Let D = {L _n } _n∈ℕ be a Lie higher derivation on 𝒯 without additive condition. Under mild assumptions, it is shown that D = {L _n } _n∈ℕ is of standard form, i.e. each component L _n (n ≥ 1) can be expressed through an additive higher derivation and a nonlinear functional vanishing on all commutators of 𝒯. As applications, nonlinear Lie higher derivations on some classical triangular algebras are characterized.

Keywords:

• nonlinear Lie higher derivation
• triangular algebra

AMS Subject Classifications::

• 15A78
• 16W25
• 47L35

Acknowledgements

The work of Z. Xiao is supported by a research foundation of Huaqiao University (Grant No. 10BS323). The work of F. Wei is partially supported by the National Natural Science Foundation of China (Grant No. 10871023). We would like to thank the referee for a very thorough reading of the manuscript and for many valuable comments. In particular, we are grateful to Professor Mikhail Chebotar's kind considerations and warm help.